Thermodynamic State Variables and Equation of State

Every equilibrium state of a thermodynamic system is completely and uniquely described by specific values of its macroscopic state variables. For a gas, this typically means its pressure $p$, volume $V$, temperature $T$, and mass $m$ (or, equivalently, the number of moles $n$). For a mixture of gases or vapours — such as the fuel-air mixture in an automobile engine — the state also needs the mixture's composition specified, in addition to $p$, $V$ and $T$.

A system's state variables are not all mutually independent; they are related to each other through a mathematical relationship called the equation of state. For an ideal gas, this is the familiar ideal gas equation:

where $p$, $V$, and $T$ are the gas's pressure, volume and temperature, $n$ is the number of moles present, and $R$ is the universal gas constant. A direct consequence of Eq. (4.5) is that for a fixed amount of gas (fixed $n$), only two of the three variables $p$, $V$, $T$ are actually independent — once any two are chosen, the equation of state fixes the third automatically. This is why, throughout the rest of the chapter, a process is typically described by specifying which pair of variables is being varied (and which one is being held fixed). …